**Purpose**

To compute (optionally) a rank-revealing RQ factorization of a real general M-by-N matrix A, which may be rank-deficient, and estimate its effective rank using incremental condition estimation. The routine uses an RQ factorization with row pivoting: P * A = R * Q, where R = [ R11 R12 ], [ 0 R22 ] with R22 defined as the largest trailing submatrix whose estimated condition number is less than 1/RCOND. The order of R22, RANK, is the effective rank of A. MB03PD does not perform any scaling of the matrix A.

SUBROUTINE MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU, $ RANK, SVAL, DWORK, INFO ) C .. Scalar Arguments .. CHARACTER JOBRQ INTEGER INFO, LDA, M, N, RANK DOUBLE PRECISION RCOND, SVLMAX C .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )

**Mode Parameters**

JOBRQ CHARACTER*1 = 'R': Perform an RQ factorization with row pivoting; = 'N': Do not perform the RQ factorization (but assume that it has been done outside).

M (input) INTEGER The number of rows of the matrix A. M >= 0. N (input) INTEGER The number of columns of the matrix A. N >= 0. A (input/output) DOUBLE PRECISION array, dimension ( LDA, N ) On entry with JOBRQ = 'R', the leading M-by-N part of this array must contain the given matrix A. On exit with JOBRQ = 'R', if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R; if M >= N, the elements on and above the (M-N)-th subdiagonal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAU, represent the orthogonal matrix Q as a product of min(M,N) elementary reflectors (see METHOD). On entry and on exit with JOBRQ = 'N', if M <= N, the upper triangle of the subarray A(1:M,N-M+1:N) must contain the M-by-M upper triangular matrix R; if M >= N, the elements on and above the (M-N)-th subdiagonal must contain the M-by-N upper trapezoidal matrix R; the remaining elements are not referenced. LDA INTEGER The leading dimension of the array A. LDA >= max(1,M). JPVT (input/output) INTEGER array, dimension ( M ) On entry with JOBRQ = 'R', if JPVT(i) <> 0, the i-th row of A is a final row, otherwise it is a free row. Before the RQ factorization of A, all final rows are permuted to the trailing positions; only the remaining free rows are moved as a result of row pivoting during the factorization. For rank determination it is preferable that all rows be free. On exit with JOBRQ = 'R', if JPVT(i) = k, then the i-th row of P*A was the k-th row of A. Array JPVT is not referenced when JOBRQ = 'N'. RCOND (input) DOUBLE PRECISION RCOND is used to determine the effective rank of A, which is defined as the order of the largest trailing triangular submatrix R22 in the RQ factorization with pivoting of A, whose estimated condition number is less than 1/RCOND. RCOND >= 0. NOTE that when SVLMAX > 0, the estimated rank could be less than that defined above (see SVLMAX). SVLMAX (input) DOUBLE PRECISION If A is a submatrix of another matrix B, and the rank decision should be related to that matrix, then SVLMAX should be an estimate of the largest singular value of B (for instance, the Frobenius norm of B). If this is not the case, the input value SVLMAX = 0 should work. SVLMAX >= 0. TAU (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) ) On exit with JOBRQ = 'R', the leading min(M,N) elements of TAU contain the scalar factors of the elementary reflectors. Array TAU is not referenced when JOBRQ = 'N'. RANK (output) INTEGER The effective (estimated) rank of A, i.e. the order of the submatrix R22. SVAL (output) DOUBLE PRECISION array, dimension ( 3 ) The estimates of some of the singular values of the triangular factor R: SVAL(1): largest singular value of R(M-RANK+1:M,N-RANK+1:N); SVAL(2): smallest singular value of R(M-RANK+1:M,N-RANK+1:N); SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N), if RANK < MIN( M, N ), or of R(M-RANK+1:M,N-RANK+1:N), otherwise. If the triangular factorization is a rank-revealing one (which will be the case if the trailing rows were well- conditioned), then SVAL(1) will also be an estimate for the largest singular value of A, and SVAL(2) and SVAL(3) will be estimates for the RANK-th and (RANK+1)-st singular values of A, respectively. By examining these values, one can confirm that the rank is well defined with respect to the chosen value of RCOND. The ratio SVAL(1)/SVAL(2) is an estimate of the condition number of R(M-RANK+1:M,N-RANK+1:N).

DWORK DOUBLE PRECISION array, dimension ( LDWORK ) where LDWORK = max( 1, 3*M ), if JOBRQ = 'R'; LDWORK = max( 1, 3*min( M, N ) ), if JOBRQ = 'N'.

INFO INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value.

The routine computes or uses an RQ factorization with row pivoting of A, P * A = R * Q, with R defined above, and then finds the largest trailing submatrix whose estimated condition number is less than 1/RCOND, taking the possible positive value of SVLMAX into account. This is performed using an adaptation of the LAPACK incremental condition estimation scheme and a slightly modified rank decision test. The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = min(m,n). Each H(i) has the form H = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in A(m-k+i,1:n-k+i-1), and tau in TAU(i). The matrix P is represented in jpvt as follows: If jpvt(j) = i then the jth row of P is the ith canonical unit vector.

[1] Bischof, C.H. and P. Tang. Generalizing Incremental Condition Estimation. LAPACK Working Notes 32, Mathematics and Computer Science Division, Argonne National Laboratory, UT, CS-91-132, May 1991. [2] Bischof, C.H. and P. Tang. Robust Incremental Condition Estimation. LAPACK Working Notes 33, Mathematics and Computer Science Division, Argonne National Laboratory, UT, CS-91-133, May 1991.

The algorithm is backward stable.

None

**Program Text**

* MB03PD EXAMPLE PROGRAM TEXT * Copyright (c) 2002-2017 NICONET e.V. * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) INTEGER NIN, NOUT PARAMETER ( NIN = 5, NOUT = 6 ) INTEGER NMAX, MMAX PARAMETER ( NMAX = 10, MMAX = 10 ) INTEGER LDA PARAMETER ( LDA = NMAX ) INTEGER LDTAU PARAMETER ( LDTAU = MIN(MMAX,NMAX) ) INTEGER LDWORK PARAMETER ( LDWORK = 3*MMAX ) * .. Local Scalars .. CHARACTER*1 JOBRQ INTEGER I, INFO, J, M, N, RANK DOUBLE PRECISION RCOND, SVAL(3), SVLMAX * .. * .. Local Arrays .. DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU) INTEGER JPVT(MMAX) * .. External Subroutines .. EXTERNAL MB03PD * .. Intrinsic Functions .. INTRINSIC MIN * .. Executable Statements .. * WRITE ( NOUT, FMT = 99999 ) * Skip the heading in the data file and read the data. READ ( NIN, FMT = '()' ) READ ( NIN, FMT = * ) M, N, JOBRQ, RCOND, SVLMAX IF ( N.LT.0 .OR. N.GT.NMAX ) THEN WRITE ( NOUT, FMT = 99972 ) N ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN WRITE ( NOUT, FMT = 99971 ) M ELSE READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M ) * RQ with row pivoting. DO 10 I = 1, M JPVT(I) = 0 10 CONTINUE CALL MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU, $ RANK, SVAL, DWORK, INFO ) * IF ( INFO.NE.0 ) THEN WRITE ( NOUT, FMT = 99998 ) INFO ELSE WRITE ( NOUT, FMT = 99995 ) RANK WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M ) WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 ) END IF END IF END IF * STOP * 99999 FORMAT (' MB03PD EXAMPLE PROGRAM RESULTS',/1X) 99998 FORMAT (' INFO on exit from MB03PD = ',I2) 99995 FORMAT (' The rank is ',I5) 99994 FORMAT (' Row permutations are ',/(20(I3,2X))) 99993 FORMAT (' SVAL vector is ',/(20(1X,F10.4))) 99972 FORMAT (/' N is out of range.',/' N = ',I5) 99971 FORMAT (/' M is out of range.',/' M = ',I5) END

MB03PD EXAMPLE PROGRAM DATA 6 5 R 5.D-16 0.0 1. 2. 6. 3. 5. -2. -1. -1. 0. -2. 5. 5. 1. 5. 1. -2. -1. -1. 0. -2. 4. 8. 4. 20. 4. -2. -1. -1. 0. -2.

MB03PD EXAMPLE PROGRAM RESULTS The rank is 4 Row permutations are 2 4 6 3 1 5 SVAL vector is 24.5744 0.9580 0.0000